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Optimal shape and location of sensors for parabolic
equations with random initial data
Yannick Privat, Emmanuel Trélat, Enrique Zuazua
To cite this version:
Yannick Privat, Emmanuel Trélat, Enrique Zuazua.
Optimal shape and location of sensors for
parabolic equations with random initial data. Archive for Rational Mechanics and Analysis, Springer
Verlag, 2015, 216 (3), pp.921–981. �hal-00965668�
Optimal shape and location of sensors for parabolic equations
with random initial data
Yannick Privat
∗Emmanuel Tr´
elat
†Enrique Zuazua
‡§Abstract
In this article, we consider parabolic equations on a bounded open connected subset Ω of IRn
. We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer?
We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of Ω of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schr¨odinger) studied in [56] for which relaxation does occur.
We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power.
The results are illustrated with several numerical simulations.
Keywords: parabolic equations, optimal design, observability, minimax theorem. AMS classification: 93B07, 35L05, 49K20, 42B37.
∗CNRS, Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005,
Paris, France ([email protected]).
†Sorbonne Universit´es, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut
Universitaire de France, F-75005, Paris, France ([email protected]).
‡BCAM - Basque Center for Applied Mathematics, Mazarredo, 14 E-48009 Bilbao-Basque Country-Spain §Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao-Basque
Contents
1 Introduction 2
2 Optimal sensor shape and location / optimal observability 8
2.1 The model. . . 8
2.2 The main result. . . 11
2.3 Application to the Stokes equation in the unit disk . . . 15
2.4 Application to anomalous diffusion equations . . . 16
2.4.1 A general result. . . 16
2.4.2 Case of the n-dimensional orthotope . . . 17
2.4.3 Case of the two-dimensional disk . . . 17
2.4.4 Several numerical simulations . . . 19
2.5 Further comments from a semi-classical analysis viewpoint. . . 21
3 Proofs 23 3.1 Proof of Proposition 2 . . . 23
3.2 Proof of Theorem 1. . . 24
3.3 Proof of Theorem 3. . . 25
3.4 Proof of Theorem 4: the n-dimensional orthotope . . . 26
3.5 Proof of Theorem 5: the unit disk of the Euclidean plane . . . 27
3.5.1 Associated radial problem . . . 28
3.5.2 Case α > 1/2 . . . 29
3.5.3 Case 0 < α < 1/2 (or α = 1/2 and T small enough). . . 30
3.6 Proof of Theorem 2. . . 43
4 Conclusion 46 4.1 Exponential concentration properties of eigenfunctions . . . 46
4.2 Several open problems . . . 47
1
Introduction
Given a bounded domain Ω of IRn, in this paper we model and solve the problem of finding an optimal observation domain ω ⊂ Ω for general parabolic equations settled on Ω. We want to optimize not only the placement but also the shape of ω, over all possible measurable subsets of Ω having a certain prescribed measure. Such questions are frequently encountered in engineering applications but have been little treated from the mathematical point of view. Our objective is here to provide a rigorous mathematical model and setting in which these questions can be addressed. Our results will be established in a general parabolic framework and cover the cases of heat equations, anomalous diffusion equations or Stokes equations. For instance for the heat equation we will answer to the following question (that we will make more precise later on):
What is the optimal shape and location of a thermometer?
Brief state of the art. Due to their relevance in engineering applications, optimal design prob-lems for the placement of sensors for processes modeled by partial differential equations have been investigated in a large number of papers. Let us mention for instance the importance of the shape and placement of sensors for transport-reaction processes (see [4, 16]). Several difficulties overlap for such problems. On the one hand, the parabolic partial differential equations under consideration constitute infinite-dimensional dynamical systems, and, consequently, solutions live
in infinite-dimensional spaces. On the other hand, the class of admissible designs is not closed for the standard and natural topology. Few works take into consideration both aspects. Indeed, in many contributions, numerical tools are developed to solve a simplified version of the optimal design problem where either the partial differential equation has been replaced with a discrete ap-proximation, or the class of optimal designs is replaced with a compact finite dimensional set (see for example [6,23,62] and [45] where such problems are investigated in a more general setting). In other words, in most of these applications the method consists in approximating appropriately the problem by selecting a finite number of possible optimal candidates and of recasting the problem as a finite-dimensional combinatorial optimization problem. In many studies the sensors have a pre-scribed shape (for instance, balls with a prepre-scribed radius) and then the problem consists of placing optimally a finite number of points (the centers of the balls) and thus it is finite-dimensional, since the class of optimal designs is replaced with a compact finite-dimensional set. Of course, the result-ing optimization problem is already challengresult-ing. We stress however that, in the present paper, we want to optimize also the shape of the observation set, and we do not make any a priori restrictive assumption to compactly the class of shapes ( ω to be of bounded variation, for instance) and the search is made over all possible measurable subsets.
From the mathematical point of view, the issue of studying a relaxed version of optimal design problems for the shape and position of sensors or actuators has been investigated in a series of articles. In [47], the authors study a homogenized version of the optimal location of controllers for the heat equation problem (for fixed initial data), noticing that such problems are often ill-posed. In [2], the authors consider a similar problem and study the asymptotic behavior as the final time T goes to infinity of the solutions of the relaxed problem; they prove that optimal designs converge to an optimal relaxed design of the corresponding two-phase optimization problem for the stationary heat equation. We also mention [46] where, for fixed initial data, numerical investigations are used to provide evidence that the optimal location of null-controllers of the heat equation problem is an ill-posed problem. In [55] we proved that, for fixed initial data as well, the problem of optimal shape and location of sensors is always well posed for heat, wave or Schr¨odinger equations (in the sense that no relaxation phenomenon occurs); we showed that the complexity of the optimal set depends on the regularity of the initial data, and in particular we proved that, even for smooth initial data, the optimal set may be of fractal type (and there is no relaxation).
A huge difference between these works and the problem addressed in this paper is that all criteria introduced in the sequel take into consideration all possible initial data. Moreover, the optimization will range over all possible measurable subsets having a given measure. This the idea developed in [53, 54, 56], where the problem of the optimal location of an observation subset ω among all possible subsets of a given measure or volume fraction of Ω was addressed and solved for conservative wave and Schr¨odinger equations. A relevant spectral criterion was introduced, viewed as a measure of eigenfunction concentration, in order to design an optimal observation or control set in an uniform way, independent of the data and solutions under consideration. Such a kind of uniform criterion was earlier introduced for the one-dimensional wave equation in [26,27] to investigate optimal stabilization issues.
The main difference of the previous analyses of conservative wave-like problems with respect to the present one is that, here, due to strong dissipativity of the heat equation (or of more general parabolic equations), high-frequency components are penalized in the spectral criterion, thus making optimal shapes to be determined by the low frequencies only, which, in particular, avoids spillover phenomena to occur.
Overview of the results of this paper. Let us now provide a short overview of the results of the present paper, without introducing (at this step) the whole general parabolic framework in which our results are actually valid.
Let Ω be an open bounded connected subset of IRn. Let T be a fixed (arbitrary) positive real number. To start with a simple model, let us consider the heat equation
∂ty − △y = 0, (t, x) ∈ (0, T ) × Ω, (1)
with Dirichlet boundary conditions. For any measurable subset ω of Ω, we observe the solutions of (1) restricted to ω over the horizon of time [0, T ], that is, we consider the observable z(t, x) = χω(x)y(t, x), where χωdenotes the characteristic function of ω. The subset ω models sensors, and
a natural question is to determine what is the best possible shape and placement of the sensors in order to maximize the observability in some appropriate sense, for instance in order to maximize the quality of the reconstruction of solutions. In other words, we ask the question of determining what is the best shape and placement of a thermometer in Ω.
At this stage, a first challenge is to settle the problem properly, to make it both mathematically meaningful and relevant in view of practical issues.
Throughout the paper, we fix a real number L ∈ (0, 1), and we will work in a class of domains ω such that |ω| = L|Ω|. In other words the set of unknowns is
UL= {χω∈ L∞(Ω; {0, 1}) | ω is a measurable subset of Ω of Lebesgue measure |ω| = L|Ω|}.
This is done to model the fact that the quantity of sensors to be employed is limited and, hence, that we cannot measure the solution over Ω in its whole.
We stress again that we do not make any restriction on the regularity or shape of the subsets ω. We are trying to determine whether or not there exists an ”absolute” optimal observation domain. We will see that such a domain exists in the parabolic case under slight assumptions on the operator and on the domain Ω (in contrast to the case of hyperbolic equations studied in [56]).
Let us now define the observability problem under consideration.
Recall that, for a given measurable subset ω of Ω, the heat equation (1) is said to be observable on ω in time T whenever there exists C > 0 such that
C Z Ω y(T, x)2dx 6 Z T 0 Z ω y(t, x)2dx dt, (2)
for every solution of (1) such that y(0, ·) ∈ D(Ω) (the set of functions defined on Ω, that are smooth and of compact support). It is well known that, if Ω is C2, then this observability inequality holds
true (see [17,20,39,61]). Note that this result has been recently extended in [5] to the case where Ω is bounded Lipschitz and locally star-shaped.
The observability constant CT(χω) is defined as the largest possible constant C > 0 such
that (2) holds. This constant gives an account for the well-posedness of the inverse problem of reconstructing the solutions from measurements over [0, T ] × ω (see, e.g., the textbook [14] for such inverse problems). Of course, the larger the constant CT(χω) is, the more stable the inverse
problem will be.
Hence it is natural to model the problem of best observation for the heat equation (1) as the problem of maximizing the functional CT(χω) over the set UL, that is,
sup
χω∈UL
CT(χω). (3)
Such a problem is however very difficult due to the presence of crossed terms at the right-hand side of (2) when considering spectral expansions (see Section2.1for details). On the other hand, actu-ally, the observability constant CT(χω) is (by nature) pessimistic in the sense that it corresponds
In practice, to reconstruct solutions one is often led to achieve a large number of measurements, and in the problem of finding a best observation domain it is reasonable to design a set that will optimize the observability only in average.
In view of that, we define an averaged version of the observability inequality, where the average runs over random initial data. This procedure, described in detail in Section 2.1, consists of randomizing the Fourier coefficients of the initial data. To explain it with few words, let us fix an orthonormal Hilbert basis (φj)j∈IN∗ of L2(Ω) consisting of eigenfunctions of the (negative of)
Dirichlet-Laplacian associated with the positive eigenvalues (λj)j∈IN∗, with λ
16· · · 6 λj→ +∞.
Every solution of (1) can be expanded as
y(t, x) =
+∞
X
j=1
aje−λjtφj(x),
We randomize the solutions (actually, their initial data) by considering
yν(t, x) = +∞ X j=1 βν jaje−λjtφj(x),
for every event ν ∈ X , where (βν
j)j∈IN∗ is a sequence of independent real random variables on a
probability space (X , A, P) having mean equal to 0, variance equal to 1, and a super exponential decay (for instance, Bernoulli laws). The randomized version of the observability inequality (2) is then defined as CT,rand(χω) Z Ω y(T, x) dx 6 E Z T 0 Z ω yν(t, x)2dx dt,
where the expectation E ranges over the space X with respect to the probability measure P. Here, CT,rand(χω) is defined as the largest possible constant such that this randomized observability
inequality holds, and is called randomized observability constant. It is easy to establish that
CT,rand(χω) = inf j∈IN∗ e2λjT − 1 2λj Z ω φj(x)2dx, (4)
for every measurable subset ω of Ω. Moreover, note that 0 6 CT,rand(χω) 6 CT(χω) (and the
second inequality may be strict, as we will see further).
Following the previous discussion, instead of considering as a criterion the deterministic observ-ability constant CT(χω) (and then, the problem (3)), we find more relevant to model the problem
of best observation domain as the problem of maximizing the functional CT,rand(χω) over the set
UL, that is the problem
sup χω∈UL CT,rand(χω) = sup χω∈UL inf j∈IN∗ e2λjT − 1 2λj Z ω φj(x)2dx. (5)
This spectral model is discussed and settled in a more general parabolic framework in Section2.1. As a particular case of our main results established in Section2.2, we have the following result for the heat equation (1) with homogeneous Dirichlet boundary conditions.
Theorem. Let T > 0 arbitrary. Assume that ∂Ω is piecewise C1. There exists a unique1 optimal
observation measurable set ω∗, solution of (5). Moreover:
1
Here, it is understood that the optimal set ω∗is unique within the class of all measurable subsets of Ω quotiented
• CT(χω∗) < CT,rand(χω∗).
• The optimal set ω∗ is open and semi-analytic. In particular, it has a finite number of con-nected components and |∂ω∗| = 0.
• The optimal set ω∗ is completely characterized from a finite-dimensional spectral
approxima-tion, by keeping only a finite number of modes. More precisely, for every N ∈ IN∗, there exists a unique measurable set ωN such that χ
ωN ∈ UL maximizes the functional
χω7−→ inf 16j6N e2λjT− 1 2λj Z ω φj(x)2dx
over UL. Moreover ωN is open and semi-analytic. Furthermore, the sequence of optimal sets
ωN is stationary, and there exists N0 ∈ IN∗ such that ωN = ω∗ for every N > N0. The
stationarity integer N0 decreases as T increases and N0= 1 whenever T is large enough. In
that case, the optimal shape is completely determined by the first eigenfunction.
A more general result (Theorem 1) will be established in a general parabolic framework. In the case of the heat equation, one of the important ingredients of the proof is a fine lower bound estimate (stated in [5]) of the spectral quantities R
ωφj(x)
2dx, which is uniform over measurable
subsets ω of a given measure.
Note that this existence and uniqueness result holds for every orthonormal basis of eigenfunc-tions of the Dirichlet-Laplacian, but the optimal set depends, in principle, on the specific choice of the basis. Of course, for T > 0 large enough, the optimal set is independent of the basis since it is completely determined by the first eigenfunction.
These properties, stated here for the heat equation (1) (and proved more generally for parabolic equations under an appropriate spectral assumption, see further) are in strong contrast with the results of [54,55,56] established for conservative wave and Schr¨odinger equations. In that context of wave-like equations it was proved that:
• when considering the problem with fixed initial data, the optimal set could be of Cantor type (hence, |∂ω| > 0) even for smooth initial data;
• the corresponding randomized observability constant is equal to infj∈IN∗
R
ωφj(x)
2dx, and,
with respect to (4), the evident difference is that all weights are equal to 1. This is not surprising in view of the conservative properties of the wave or Schr¨odinger equation, however the fact that all frequencies have the same weight causes a strong instability of the optimal sets ωN (maximizers of the corresponding spectral approximation). It was proved in [27,54]
that the best possible set ωN for N modes is actually the worst possible one when considering
N + 1 modes (spillover phenomenon).
In contrast, for the parabolic problems under consideration, we prove that this instability phenomenon does not occur, and that the sequence of maximizers ωN is constant for N large
enough, equal to the optimal set ω∗. This stationarity property is of particular interest in view of
designing the best observation set ω∗in practice.
In Section 2.2 we provide more details on these results, and state them in a far more gen-eral setting, involving in particular the Stokes equation and anomalous diffusion equations (with fractional Laplacian). For the Stokes equation
∂ty − △y + ∇p = 0, div y = 0, (6)
considered on the unit disk with Dirichlet boundary conditions, we establish that there exists a unique optimal observation set in UL, sharing nice regularity properties as above.
Let us mention a striking feature occuring for the anomalous diffusion equation
∂ty + (−△)αy = 0, (7)
considered on some domain Ω, where (−△)αis some positive power of the Dirichlet-Laplacian. Note
that such equations are well recognized as being relevant models in many problems encountered in physics (plasma with slow or fast diffusion, aperiodic crystals, spins, etc), in biomathematics, in economy, also in imaging sciences (see for instance [42,44,60]). Hence they provide an important class of parabolic equations entering into the general framework developed in the paper.
Given T > 0 arbitrary, we prove that if ∂Ω is piecewise C1 and if α > 1/2 (or if α = 1/2 and
T is large enough) then there exists a unique optimal observation domain, independently on the Hilbert basis of eigenfunctions under consideration. Furthermore, we prove the unexpected facts that:
• in the Euclidean square Ω = (0, π)2, when considering the usual Hilbert basis of
eigenfunc-tions consisting of products of sine funceigenfunc-tions, for every α > 0 there exists a unique optimal set in UL(as in the theorem), which is moreover open and semi-analytic and thus has a finite
number of connected components (and this, whatever the value of α > 0 may be);
• in the Euclidean disk Ω = {x ∈ IR2 | kxk < 1}, when considering the usual Hilbert basis of eigenfunctions parametrized in terms of Bessel functions, for every α > 0 there exists a unique optimal set ω∗(as in the theorem), which is moreover open, radial, with the following
additional property:
– if α > 1/2 then ω∗ consists of a finite number of concentric rings that are at a positive
distance from the boundary;
– if α < 1/2 (or if α = 1/2 and T is small enough) then ω∗ consists of an infinite number
of concentric rings accumulating at the boundary!
This surprising result shows that the complexity of the optimal shape does not only depend on the operator but also on the geometry of the domain Ω.
It must be underlined that the proof of these properties (done in Section 3.5) is lengthy and particularly difficult in the case α < 1/2. It requires the development of very fine estimates for Bessel functions, combined with the use of quantum limits (semi-classical measures) in the disk, nontrivial minimax arguments and analyticity considerations.
Several numerical simulations based on the spectral approximation described previously are provided in Section 2.4. They show in particular what is the optimal shape and location of a thermometer in a square or in a disk.
The paper is structured as follows.
Section 2is devoted to model and solve the problem of finding a best observation domain for parabolic equations. The model is discussed and defined in Section2.1, based on the introduction of the randomized observability inequality. The problem is solved in a general parabolic setting in Section2.2, where it is shown that, under an appropriate spectral assumption, there exists a unique optimal observation set, which can moreover be recovered from a finite dimensional spectral approximation problem. Section2.3 is devoted to the application to the Stokes equation on the unit disk. In Section2.4, we study the case of anomalous diffusion equations and then we provide several numerical simulations illustrating our results and in particular the stationarity feature of the sequence of optimal sets. Further comments on the spectral assumption are presented in Section2.5, from a semi-classical analysis viewpoint.
All results are proved in Section 3. It must be underlined that the proof concerning the anomalous diffusion equations, in particular in the case α < 1/2, is long and very technical. It
is actually unexpectedly difficult. The proof concerning the Stokes equation is as well for a large part based on facts derived in the previous proof.
Section4 provides a conclusion and several further comments and open problems.
2
Optimal sensor shape and location / optimal observability
Let Ω be an open bounded connected subset of IRn. Throughout the paper we consider the problem of determining the optimal observation domain for the abstract parabolic model
∂ty + A0y = 0, (8)
where A0 : D(A0) → L2(Ω, C) be a densely defined operator. Precise assumptions on A0 will be
done further. As the main reference, we can keep in mind the typical example of the heat equation with Dirichlet boundary conditions overviewed in the introduction. But our analysis and results will be established for a large class of parabolic operators.
At this stage all what we need to assume, in order to establish the model that we will study, is that there exists a normalized Hilbert basis (φj)j∈IN∗ of L2(Ω, C) consisting of (complex-valued)
eigenfunctions of A0, associated with the (complex) eigenvalues (λj)j∈IN∗.
2.1
The model
The aim of this section is to introduce and define a relevant mathematical model of the problem of best observation. The first ingredient is the notion of observability inequality.
Observability inequality. For every y0∈ D(A
0), there exists a unique solution y ∈ C0(0, T ; D(A0))∩
C1(0, T ; L2(Ω)) of (8) such that y(0, ·) = y0(·). For every measurable subset ω of Ω, the equation
(8) is said to be observable on ω in time T if there exists C > 0 such that
Cky(T, ·)k2L2(Ω)6 Z T 0 Z ω|y(t, x)| 2dx dt, (9)
for every solution of (8) such that y(0, ·) ∈ D(A0). This inequality is called observability inequality,
and the constant defined by
CT(χω) = inf ( RT 0 R ω|y(t, x)| 2dx dt ky(T, ·)k2 L2(Ω) y0∈ D(A0) \ {0} ) , (10)
is called the observability constant. It is the largest possible nonnegative constant for which (9) holds. In other words, the equation (8) is observable on ω in time T if and only if CT(χω) > 0.
Remark 1. It is well known that, if A0 is the negative of the Dirichlet, or Neumann, or Robin
Laplacian, then the equation (8) is observable (see [17, 20, 39, 61]), for every open subset ω of Ω. The observability property holds as well, e.g., for the linearized Cahn-Hilliard operator corresponding to Ω ⊂ IRn, A0 = (−△)2, with the boundary conditions y|∂Ω = △y|∂Ω = 0 (see
[61]). For the Stokes operator, the observability property follows from [19, Lemma 1].2
2
More precisely, in order to derive the usual observability inequality from the Carleman estimate proved in this reference, it suffices to estimate from below the left-hand side weight on [T /4, 3T /4], to estimate from above the right-hand weight, and to use the fact that the function t 7→ ky(t, ·)kL2 is nonincreasing.
As explained in the introduction, throughout the paper we fix a real number L ∈ (0, 1) and we will search an optimal domain in the set
UL= {χω∈ L∞(Ω; {0, 1}) | ω is a measurable subset of Ω of Lebesgue measure |ω| = L|Ω|}. (11)
This gives an account for the fact that we can measure the solutions only over a part of the whole domain Ω.
Having in mind the observability inequality (9), it is a priori natural to model the question of the optimal location of sensors in terms of maximizing the observability constant CT(χω) over the set
UL defined by (11), where T > 0 is fixed. Actually, when implementing a reconstruction method,
the observability constant CT(χω) gives an account for the well-posedness of the corresponding
inverse problem. More precisely, the larger the observability constant is, and the better conditioned the inverse problem is.
However at this stage two remarks are in order.
Firstly, settled as such, the problem is difficult to handle, due to the presence of crossed terms at the right-and side of (9) when considering spectral expansions. This problem, which has been discussed thoroughly in [54, 56], is quite similar to the open problem of determining the best constants in Ingham’s inequalities (see [29,30]). Here, one is faced with the problem of determining the infimum of eigenvalues of an infinite dimensional symmetric nonnegative matrix (namely, the Gramian, see below). Although this criterion has a clear sense, it leads to an optimal design problem which does not seem to be easily tractable.
Secondly, even though the problem of maximizing the observability constant seems natural at the first glance, it is actually not so relevant with respect to the practical issues that we have in mind. Indeed in practice one is led to deal with a large number of solutions: when implementing a reconstruction process, one has to carry out in general a very large number of measures; likewise, when implementing a control procedure, the control strategy is expected to be efficient in general, but maybe not exactly for all cases. The issue that we raise here is the fact that the above observability inequality (9) is deterministic, and thus the observability constant CT(χω)
is pessimistic since it corresponds to a worst possible case. It is likely that in practice this worst case will not occur very often, and hence the deterministic observability constant is not a relevant criterion when realizing a large number of experiments. Instead of that, we are going to propose an averaged version of the observability constant, better suited to our purposes, and defined in terms of probabilistic arguments.
We next describe this procedure, inspired by [11] and which has been used as well in [56] to deal with wave and Schr¨odinger equations.
Randomized observability inequality. Every solution y(·) of (8) such that y(0, ·) = y0(·) can
be expanded as y(t, x) = +∞ X j=1 aje−λjtφj(x), (12) where aj= Z Ω y0(x)φ j(x) dx, (13)
for every j ∈ IN∗. Using this spectral decomposition, the change of variable bj = aje−λjT and an
easy density argument, we get
CT(χω) =P inf +∞ j=1|bj|2=1 Z T 0 Z ω +∞ X j=1 bjeλjtφj(x) 2 dx dt. (14)
As briefly explained previously, CT(χω) appears as the infimum of the eigenvalues of a Gramian
operator, which is the infinite-dimensional Hermitian nonnegative matrix
GT(χω) = e(λj+¯λk)T − 1 λj+ ¯λk Z ω φj(x)φk(x) dx ! j,k>1 . (15)
Due to the crossed terms appearing when expanding the square in (14), the resulting optimal design problem, consisting of maximizing CT(χω) over the set UL, is not easily tractable, at least
in view of deriving theoretical results. Moreover, from the practical point of view the problem of modeling the best observation has to be done, having in mind that the best observation domain should be designed to be the best possible in average, that is, over a large number of experiments. The observability constant CT(χω) above is by definition deterministic, and thus pessimistic in the
sense that is gives an account for the worst possible case. In practice, when carrying out a large number of experiments, it can however be expected that the worst possible case does not occur very often. Having this remark in mind, we next define a new notion of observability inequality by considering an average over random initial data. We then define below a notion of randomized
observability constant, which is in our view better suited to the model of best observation. We
follow [56], accordingly to early ideas developed in [50] for harmonic analysis issues and recently in [10, 11] in view of ensuring the probabilistic well-posedness of classically ill-posed supercritical wave or Schr¨odinger equations.
For any given y0 ∈ D(A
0), the Fourier coefficients of y0, defined by (13), are randomized by
defining aν
j = βjνaj for every j ∈ IN∗, where (βνj)j∈IN∗ is a sequence of independent real random
variables on a probability space (X , F, P) having mean equal to 0, variance equal to 1, and a super exponential decay (for instance, independent Bernoulli random variables, see [10, 11] for more details on randomization possibilities and properties). For every ν ∈ X , the solution corresponding to the initial data y0
ν =
P+∞
j=1βjνajφjis then yν(t, ·) =Pj=1+∞βνjaje−λjtφj(·). Instead of considering
the deterministic observability inequality (9), we define the randomized observability inequality by
CT,rand(χω)ky(T, ·)k2L2(Ω) 6 E Z T 0 Z ω|y ν(t, x)|2dx dt, (16)
for every solution y of (8) such that y(0) ∈ D(A0), where E is the expectation over the space X with
respect to the probability measure P. The nonnegative constant CT,rand(χω) is called randomized
observability constant and is defined (by density) by
CT,rand(χω) =P inf +∞ j=1|bj|2=1 E Z T 0 Z ω +∞ X j=1 βjνbjeλjtφj(x) 2 dx dt. (17)
It is the randomized counterpart of the deterministic constant CT(χω) defined by (14). Note that
0 6 CT(χω) 6 CT,rand(χω), (18)
for every measurable subset ω of Ω. The inequalities can be strict (see Theorem1further). Proposition 1. Let T > 0 arbitrary. For every measurable subset ω of Ω, we have
CT,rand(χω) = inf j∈IN∗γj(T ) Z ω|φ j(x)|2dx, with γj(T ) = e2Re(λj)T − 1 2Re(λj) if Re(λj) 6= 0, T if Re(λj) = 0. (19)
Proof. Using the Fubini theorem and the independence of the random laws, one has CT,rand(χω) = P inf +∞ j=1|bj|2=1 Z T 0 Z ω +∞ X j,k=1 E(βν jβjν)bj¯bke(λj+¯λk)tφj(x)φk(x) dx dt = P inf +∞ j=1|bj|2=1 +∞ X j=1 |bj|2 Z T 0 e2Re(λj)tdt Z ω|φ j(x)|2dx,
and the conclusion follows easily.
This result clearly shows how the randomization procedure rules out the off-diagonal terms in the Gramian (15).
Conclusion: the optimal shape design problem. For every measurable subset ω of Ω, we set J(χω) = CT,rand(χω) = inf j∈IN∗γj(T ) Z ω|φ j(x)|2dx, (20)
Throughout the paper, we will consider the problem of maximizing the functional J over the set UL defined by (11), where the coefficients γj(T ) are defined by (19). In other words, we consider
the problem sup χω∈UL J(χω) = sup χω∈UL inf j∈IN∗γj(T ) Z ω|φ j(x)|2dx. (21)
According to the previous discussion, this optimal shape design problem models the best sensor shape and location problem for the parabolic equation (8).
The functional J defined by (20) corresponds to an energy concentration measure. As we will see, solving this problem requires spectral assumptions.
2.2
The main result
In our main result below, it will be useful to consider the functional JN defined by
JN(χω) = inf
16j6Nγj(T )
Z
ω|φ
j(x)|2dx, (22)
for every measurable subset ω of Ω, for every N ∈ IN∗. The functional JN is the spectral truncation
of the functional J to the N first terms. We consider as well the shape optimization problem sup
χω∈UL
JN(χω), (23)
which is a spectral approximation of the problem (21). We call it the truncated problem. Let us now provide the general parabolic framework and the required spectral assumptions.
Framework and assumptions. Let Ω be an open bounded connected subset of IRn, and let L ∈ (0, 1) and T > 0 be arbitrary. Let A0 : D(A0) → L2(Ω, C) be a densely defined operator,
generating a strongly continuous semigroup on L2(Ω, C). We assume that there exists a Hilbert
basis (φj)j∈IN∗ of L2(Ω, C) consisting of (complex-valued) eigenfunctions of A
0, associated with
(complex) eigenvalues (λj)j∈IN∗ such that Re(λ
1) 6 · · · 6 Re(λj) 6 · · · , and such that the following
(H1) (Strong Conic Independence Property) If there exists a subset E of Ω of positive Lebesgue
measure, an integer N ∈ IN∗, a N -tuple (αj)16j6N ∈ (IR+)N, and C > 0 such that
PN
j=1αj|φj(x)|2 = C almost everywhere on E, then there must hold C = 0 and αj = 0
for every j ∈ {1, · · · , N}.
(H2) For every a ∈ L∞(Ω; [0, 1]) such that
R
Ωa(x) dx = L|Ω|, one has
lim inf
j→+∞ γj(T )
Z
Ωa(x)|φ
j(x)|2dx > γ1(T );
(H3) The eigenfunctions φj are analytic in Ω.
We start with a simple preliminary result for the truncated problem.
Proposition 2. Under (H1), for every N ∈ IN∗, the truncated problem (23) has a unique3solution
χωN ∈ UL. Moreover, under (H3), ωN is an open semi-analytic4 set, and thus, in particular, it
has a finite number of connected components.
Remark 2. If A0 is defined on a domain D(A0) such that the eigenfunctions φj vanish on ∂Ω
(Dirichlet boundary conditions), then moreover there exists ηN > 0 such that the (Euclidean)
distance between ωN and ∂Ω is larger than ηN.
Our main result is the following.
Theorem 1. Under (H1) and (H2), the optimal shape design problem (21) has a unique solution
χω∗ ∈ U
L.
Moreover, there exists a smallest integer N0(T ) such that
J(χω∗) = max
χω∈UL
J(χω) = max χω∈UL
JN(χω),
for every N > N0(T ). In other words, the sequence (χωN)N ∈IN∗ of maximizers of JN is stationary,
that is, ω∗= ωN0(T )= ωN for N > N
0(T ).
The function T 7→ N0(T ) is nonincreasing, and if Re(λj) → +∞ as j → +∞ then N0(T ) = 1
whenever T is large enough.
Under the additional assumption (H3), we have moreover that:
• CT(χω∗) < CT,rand(χω∗);
• the optimal observation set ω∗ is an open semi-analytic set and thus it has a finite number
of connected components.
Proposition2 and Theorem1are proved in Section 3.
Remark 3. In the next sections we will comment in detail on the assumptions done in the theorem, and provide classes of examples where they are satisfied (note however that proving their validity is far from obvious): heat and anomalous diffusion equations, Stokes equation.
We can however note, at this stage, that these assumptions are of different natures.
3
Here and in the sequel, it is understood that the optimal set is unique within the class of all measurable subsets of Ω quotiented by the set of all measurable subsets of Ω of zero measure.
4
A subset ω of a real analytic finite dimensional manifold M is said to be semi-analytic if it can be written in terms of equalities and inequalities of analytic functions. We recall that such semi-analytic subsets are stratifiable in the sense of Whitney (see [22,28]), and enjoy local finitetess properties, such that: local finite perimter, local finite number of connected components, etc.
The assumption (H1) will be treated essentially with analyticity considerations. Indeed note
that (H1) holds true as soon as the eigenfunctions φj are analytic in Ω (that is, under the
assump-tion (H3)) and vanish along ∂Ω. This is often the case, for instance, for elliptic operators with
analytic coefficients. It can be noted that a generalization of the property (H1) has been studied
for the Dirichlet-Laplacian in [52], where the αj are arbitrary real numbers, and is proved to hold
generically with respect to the domain Ω. The validity of (H1) in general (for instance, in for
Neumann boundary conditions) is an open problem.
The assumption (H2), which can as well be seen from a semi-classical point of view (see
comments in Section 2.5 further) is related with nonconcentration properties of eigenfunctions. For instance proving it for heat-like equations will require the use of fine recent results providing lower bound estimates that are uniform with respect to the observation domain ω.
Before coming to these applications, several remarks are in order.
Remark 4. The fact that the sequence (χωN)N ∈IN∗ of optimal sets of the truncated problem (35)
is stationary is in strong contrast with the results of [26, 27, 53, 54, 56] in which such optimal design problems have been investigated for conservative wave or Schr¨odinger equations. In these references it was observed and proved that the corresponding maximizing sequence of subsets does not converge in general, except in very particular cases. Moreover, in dimension one, this sequence of sets has an instability property known as spillover phenomenon. Namely, the best possible set for N modes is actually the worst possible one when considering N + 1 modes. This instability property has negative consequences in view of practical issues for designing a relevant notion of optimal set.
In contrast, Theorem1shows that, for the parabolic equation (8), the maximizing sequence of subsets is stationary, and hence only a finite number of modes is enough in order to capture all the information necessary to design the true optimal set. In other words, higher modes play no role. Although this result can appear as intuitive because we are dealing with a parabolic equation, deriving such a property however requires the spectral property (H2), which is commented and
analyzed further.
Remark 5. The fact that the optimal set ω∗is semi-analytic is a strong (and desirable) regularity
property. In addition to the fact that ω∗has a finite number of connected components, this implies
also that ω∗is Jordan measurable, that is, |∂ω∗| = 0. This is in contrast with the already mentioned
fact that, for wave-like equations, when maximizing the energy for fixed data, the optimal set may be a Cantor set of positive measure, even for smooth initial data (see [55]).
Remark 6 (A convexified formulation of (21)). It is standard in shape optimization to introduce a convexified version of a maximization problem, since it may fail to have some solutions because of hard constraints. This is what is usually referred to as relaxation (see, e.g., [9]).
Since the set UL(defined by (11)) does not share nice compactness properties, we consider the
convex closure of UL for the weak star topology of L∞, which is
UL= a ∈ L∞(Ω; [0, 1]) | Z Ωa(x) dx = L|Ω| . (24)
Such a relaxation was used as well in [47, 54, 56]. Replacing χω ∈ UL with a ∈ UL, we define a
relaxed formulation of the optimal shape design problem (21) by sup
a∈UL
J(a), (25)
where the functional J is naturally extended toUL by
J(a) = inf
j∈IN∗γj(T )
Z
Ωa(x)|φ
for every a ∈ UL. Moreover, one has the following existence result.
Lemma 1. For every L ∈ (0, 1), the relaxed problem (25) has at least one solution a∗∈ U L.
Proof of Lemma1. For every j ∈ IN∗, the functional a ∈ UL7→ γjRΩa(x)|φj(x)|2dx is linear and
continuous for the weak star topology of L∞. Hence J is upper semicontinuous as the infimum of
continuous linear functionals. SinceUL is compact for the weak star topology of L∞, the lemma
follows.
Note that, obviously,
sup
χω∈UL
J(χω) 6 sup a∈UL
J(a) = J(a∗).
But, in fact, from Theorem1 (and from its proof) we deduce that the two suprema coincide, and that the problem (21) and the relaxed problem (25) have the same (unique) solution. This means two things. First, there is no gap between the optimal values of the problem (21) and its relaxed formulation (25). A similar result was established in [56] for wave and Schr¨odinger like equations under spectral assumptions on the domain Ω. But, in contrast to these hyperbolic equations where relaxation occurs except for some very distinguished discrete values of L, here, in the parabolic setting, relaxation does not occur, at least under the assumption (H2), which is fulfilled for the
Dirichlet-Laplacian for piecewise C1 domains Ω (see Theorem3further).
In particular, in the parabolic setting, contrarily to what happens in wave-like equations, the constant function a = L is not an optimal solution. Note that this constant function corresponds intuitively (at the weak limit) to equi-distribute the sensors over the domain Ω. This strategy is however not optimal for parabolic problems.
Remark 7. The assumption (H2) can be actually weakened (as can be easily seen from the proof
of the theorem). To ensure that the conclusion of Theorem1 holds, it is sufficient to assume that
lim inf
j→+∞ γj(T )
Z
Ω
a∗(x)|φj(x)|2dx > γ1(T ), (27)
where a∗∈ ULis any optimal solution of the relaxed problem (25). In other words, it is sufficient
to restrict the assumption (H2) to the sole a∗. Note that such an assumption is impossible to
check since a∗ is not known a priori, but this remark will however be useful in Section 3.5.
Note that, since J(L) = Lγ1, it follows that J(a∗) > Lγ1, and hence in particular there always
holds lim inf j→+∞ γj(T ) Z Ω a∗(x)|φj(x)|2dx > Lγ1(T ).
Remark 8. The existence and uniqueness of an optimal set, stated in Theorem1, holds true for any Hilbert basis of eigenfunctions of A0as soon as this basis satisfies the assumptions (H1), (H2)
and (H3). However the optimal set ω∗ may depend on the specific choice of the basis.
Remark 9. As noted before, the issue of solving the optimal design problem sup
χω∈UL
CT(χω)
where CT(χω) is the observability constant of the parabolic equation (8) defined by (10), is natural
and interesting, although this problem is very difficult to handle from the theoretical point of view, even for the truncated criterion, and not as much relevant as the one we consider here, from the practical point of view (as already discussed).
Note that the truncated version of the criterion CT,rand(χω) is the lowest eigenvalue of the
diagonal matrix diag γj(T )Rω|φj(x)|2dx16j6N, whereas the truncated version CT,N(χω) of the
criterion CT(χω) is the lowest eigenvalue of the Gramian matrix
GT,N(χω) = e(λj+¯λk)T− 1 λj+ ¯λk Z ω φj(x)φk(x) dx ! 16j,k6N , (28)
which is the truncation of the Gramian GT(χω) defined by (15). Under the conditions of Theorem 1, the sequence of the minimizers over UL of the truncated version of the randomized constant
CT,rand(χω) is stationary. An interesting problem consists of investigating theoretically or
numer-ically whether this stationarity property holds true or not for the truncated version CT,N(χω) of
the observability constant CT(χω).
Notice that, extending the definition of CT(χω) to the functions a ∈ L∞(Ω; [0, 1]) by
CT(a) = inf ( RT 0 R Ωa(x)|y(t, x)| 2dx dt ky(T, ·)k2 L2(Ω) y0∈ D(A0) \ {0} ) ,
one gets easily that the optimal design problem of maximizing CT(a) over UL has at least one
solution. Furthermore, it is interesting to note that, by adapting the proof of [54, Proposition 2], we get the following partial result.
Lemma 2. For every L ∈ (0, 1) and every T > 0, the constant function a(·) = L is not a maximizer
of the functional a 7→ CT(a) overUL.
Remark 10. Finally, let us comment on the role of the time T . Recall that T > 0 has been arbitrarily fixed at the beginning of the analysis. Its role is in the weights γj(T ) coming into
play in the definition of the functional J (defined by (20)). If the eigenvalues are such that Re(λj) → +∞, then the larger T is, and the quicker the weights tend to +∞. As a consequence,
as stated in Theorem1, the integer N0(T ) decreases as T increases, and if T is large enough then
N0(T ) = 1. This says that, if one can observe the solutions of the equation over a large enough
horizon of time, then the optimal observation domain can be designed from the first mode only. This fact is in accordance with the strong damping properties of a parabolic equation, at least, under the assumption (H2). In large time the energy of the solutions is essentially carried by the
first mode.
2.3
Application to the Stokes equation in the unit disk
In this section, we assume that Ω = {x ∈ IR2 | kxk < 1} is the Euclidean unit disk of IR2, and we consider the Stokes equation (6) in the unit disk of IR2, with Dirichlet boundary conditions.
Note that the Stokes system does not exactly enter in the framework defined in Section2.2, but it suffices to make the following very slight modification. The Stokes operator A0 : D(A0) → H
is defined by A0 = −P△, with D(A0) = {y ∈ V | A0y ∈ H}, V = {y ∈ (H01(Ω))2 | div y = 0},
H = {y ∈ (L2(Ω))2 | div y = 0, y
|∂Ω.n = 0}, and P : (L2(Ω))2 = H ⊥
⊕ H⊥ → H is the Leray
projection. Then A0 is an unbounded operator in the Hilbert space H (and not on L2), endowed
with the L2-norm (see [8]).
We consider here the Hilbert basis of H of eigenfunctions, indexed by j ∈ Z, k ∈ IN∗ and m = 1, 2, defined by φ0,k(r, θ) = −J ′ 0(pλ0,kr) √πpλ 0,k|J0(pλ0,k)| − sin θ cos θ , (29)
and φj,k,m(r, θ) = Jj(pλj,kr) − Jj(pλj,k)rj λj,k|Jj(pλj,k)|r j(−1) m+1Y j,m(θ) cos θ sin θ +−pλj,kJ ′ j(pλj,kr) + jJj(pλj,k)rj−1 λj,k|Jj(pλj,k)| Yj,m+1(θ)− sin θ cos θ (30)
whenever j 6= 0, where (r, θ) are the usual polar coordinates (see [34,40]). The functions Yj,m(θ)
are defined by Yj,1(θ) = √1πcos(jθ) and Yj,2(θ) = √1πsin(jθ), with the agreement that Yj,3 = Yj,1,
and Jjis the Bessel function of the first kind of order j. Denoting by zj,k> 0 is the kthpositive zero
of Jj, the eigenvalues of A0are the doubly indexed sequence (−λj,k)j∈Z,k∈IN∗, where λ
j,k= z2|j|+1,k
is of multiplicity 1 if j = 0, and 2 if j 6= 0.
Note that, in (H1), (H2), and in the definition (20) of the functional J, we replace | · | with
the Euclidean norm of IR2.
Theorem 2. The assumptions (H1), (H2) and (H3) are satisfied. Then Theorem 1 implies
that there exists a unique optimal observation domain ω∗ (solution of the problem (21)), which is
moreover open and semi-analytic.
This result is proved in Section3.6. The proof is technically based on the explicit form of the basis of eigenfunctions under consideration, and we did not investigate what can happen in higher dimension. Also, what can happen for more general domains is not known.
2.4
Application to anomalous diffusion equations
In this section, we assume that Ω is Lipschitz, and we consider the Dirichlet-Laplacian △ defined on its domain D(△) = {y ∈ H1
0(Ω) | △y ∈ L2(Ω)}. Note that if ∂Ω is C2then D(△) = H01(Ω)∩H2(Ω).
We set A0 = (−△)α (where △ is the Dirichlet-Laplacian), with α > 0 arbitrary, defined
spectrally, based on the spectral decomposition of the Dirichlet-Laplacian. This case corresponds to the anomalous diffusion equation (7).
To be more precise with the functional framework, the domain of the operator A0, as an
unbounded operator in L2(Ω), is defined as follows. If α ∈ (0, 1) \ {1/4}, then D(A
0) = H02α(Ω);
if α = 1/4 then D(A0) = H001/2(Ω) (Lions-Magenes space), and if 1/4 < α < 1 then D(A0) =
H1
0(Ω) ∩ H2s(Ω) (see [41] or [7, Appendix]).
For α > 1 the operator is defined by composing integer powers of −△ with the fractional powers above. For instance one can take A0 = (−△)2 with the boundary conditions y|∂Ω = △y|∂Ω = 0:
in that case (8) corresponds to a linearized model of Cahn-Hilliard type.
In the general case α > 0, the equation (8) models a physical process exhibiting anomalous diffusion (see for instance [42, 44, 60]). Of course if α = 1 then (8) is the heat equation with Dirichlet boundary conditions, as overviewed in the introduction.
Note that the eigenfunctions of A0 are those of the Dirichlet-Laplacian, and therefore the
assumptions (H1) and (H3) are satisfied. Only the assumption (H2) has to be discussed in the
sequel. We have the following three results. 2.4.1 A general result
Theorem 3. Assume that ∂Ω is piecewise5 C1. If α > 1/2, then the assumption (H
2) is satisfied
for any Hilbert basis of eigenfunctions of A0. For α = 1/2, the conclusion holds true as well
provided that T is moreover large enough. 5
Actually a more general assumption can be done: Ω is Lipschitz and locally star-shaped (see [5] for the definition and details).
Under these conditions, Theorem1can be applied and implies that there exists a unique optimal set ω∗ (solution of the problem (21)), which is open and semi-analytic.
In order to prove that the uniform lower bound assumption (H2) holds true, the main ingredient
is a lower bound estimate (stated in [5]) of the spectral quantities R
ωφj(x)
2dx, which is uniform
over measurable subsets ω of a given measure.
It can be noted that the number N0(T ) of relevant modes needed to compute the optimal set
depends on the speed of convergence of j2α−1n T to +∞ (this follows from the proof of Theorem3,
by using Weyl’s asymptotics).
2.4.2 Case of the n-dimensional orthotope
Assume that Ω = (0, π)n, for n ∈ IN∗. We consider the usual Hilbert basis consisting of products
of sine eigenfunctions, given by
φj1,...,jn(x) = 2 π n/2 n Y k=1 sin(jkxk), (31)
for all (j1, . . . , jn) ∈ IN∗n, with corresponding eigenvalues λ(j1,...,jn)=
Pn
k=1jk2
α .
Theorem 4. The assumption (H2) is satisfied, whatever the value of α > 0 may be. Then,
Theorem1implies that there exists a unique optimal observation domain ω∗(solution of the problem
(21)), which is open and semi-analytic.
Note that it is not clear whether this result is satisfied or not for any Hilbert basis of eigen-functions, at least for α < 1/2 (indeed the case α > 1/2 is solved with Theorem3).
As shown in the proof of Theorem 4, it can be also noted that the conclusion of Theorem 1
holds with N0(T ) defined as the lowest multi-index (j1, . . . , jn) (in lexicographical order) such that
e2λ(j1,...,jn)T − 1 2λ(j1,...,jn) > e 2λ(1,...,1)T − 1 2λ(1,...,1)F[n](Lπn) .
where F is the function defined on [0, π] by F (s) =π1(s − sin s), and F[n] is the composition of F
with itself, n times.
Remark 11. Note that the result of Theorem 4 holds true as well for the Neumann-Laplacian A0= −△ defined on the domain D(A0) = {y ∈ H2(Ω, C) | RΩy = 0 and ∂y∂n = 0 on ∂Ω}, with the
usual Hilbert basis of eigenfunctions consisting of products of cosine functions (it is indeed easy to see that the assumption (H2) is satisfied). Fractional operators can be as well defined out of this
Neumann-Laplacian. The reason to consider the Neumann-Laplacian on functions that are of zero average (which is standard for observability issues) is due to the fact that, if we do not make this restriction then λ1= 0 (and λj> 0 for every j > 2) and φ1= 1/p|Ω|, and (H1) fails.
2.4.3 Case of the two-dimensional disk
Assume that Ω = {x ∈ IR2 | kxk < 1} is the Euclidean unit disk of IR2. We consider the Hilbert basis of eigenfunctions defined by the triply indexed sequence of functions
φj,k,m(r, θ) = R0,k(r)/ √ 2π if j = 0, Rj,k(r)Yj,m(θ) if j > 1, (32)
for j ∈ IN, k ∈ IN∗ and m = 1, 2, where (r, θ) are the usual polar coordinates. The functions Yj,m(θ) are defined by Yj,1(θ) = √1πcos(jθ) and Yj,2(θ) = √1πsin(jθ), and the functions Rj,k are
defined by Rj,k(r) = √ 2Jj(zj,kr) |J′ j(zj,k)| =q Jj(zj,kr) R1 0 Jj(zj,kr)2r dr , (33)
where Jj is the Bessel function of the first kind of order j, and zj,k> 0 is the kth positive zero of
Jj. The corresponding eigenvalues consist of a doubly indexed sequence (−λj,k)j∈IN,k∈IN∗, where
λj,k= zj,k2α is of multiplicity 1 if j = 0, and 2 if j > 1.
Theorem 5. For every α > 0, the optimal design problem (21) has a unique solution χω∗ ∈ U
L,
where ω∗ is moreover open and radial. Furthermore:
• If α > 1/2 then the assumption (H2) is satisfied6 and ω∗ consists of a finite number of
concentric rings that are at a positive distance from the boundary. Additionally, we have
limj→+∞Rω∗φj,k,m(x)
2dx = 0, for every k ∈ IN∗.
• If 0 < α < 1/2, or if α = 1/2 and T is small enough, then neither the assumption (H2) nor
its weakened version (27) are satisfied. The optimal set ω∗ consists of an infinite number of
concentric rings accumulating at the boundary. However the number of connected components of ω∗ intersected with any proper compact subset of Ω is finite.
Theorems3, 4and5 are proved in Section3.
Theorem5 is probably the most difficult result of the paper. Its contents contrast with those of Theorem4. Indeed, for instance in the two-dimensional square, the optimal observation domain consists of a finite number of connected components, which are at a positive distance from the boundary (since we are in the Dirichlet case), and this whatever the value of α > 0 may be. In the two-dimensional disk, we have a similar conclusion for α > 1/2, but if α < 1/2 then the optimal domain is much more complex and has an infinite number of connected components. This surprising result shows that the complexity of the optimal domain ω∗ depends on the geometry of
the whole Ω.
Note that, in Theorem 5, the result of Theorem 1 cannot be applied if α < 1/2. We are however able to prove the existence and the uniqueness of an optimal domain. The proof relies on a nontrivial minimax argument combined with fine properties of Bessel functions, analyticity considerations, and the use of quantum limits in the disk.
Remark 12. For α > 1/2, the fact that lim infj+k→+∞R
ω∗φj,k,m(x)
2dx = 0 in spite of (H 2) is
in contrast with the results given in Theorems3and4 where this limit was positive.
At this step the role of the weights γj,k(T, α) must be underlined. Indeed, in the disk there is the
well-known whispering gallery phenomenon, according to which a subsequence of the probability measures φ2
j,k,mdx converges vaguely to the Dirac along the boundary (this property is recalled in
a precise way in the proof of Lemma17in terms of semi-classical limits). Note that the whispering gallery concentration phenomenon is however not strong enough to imply the failure of (H2) if
α > 1/2, due to the exponential increase of the coefficients γj,k(T, α) as j + k tends to +∞ (see
also Section2.5).
In contrast, if α ∈ (0, 1/2) then the increase of the coefficients γj,k(T, α) is not strong enough,
which is in accordance with the fact that (H2) fails.
6
Remark 13. As noted in the inequality (18), there holds CT(χω) 6 CT,rand(χω), for every
mea-surable subset ω of Ω. The last part of Theorem1states that the inequality is strict for the optimal set ω∗. Combining this remark with Theorem4, it is interesting to note the following fact.
Assume that Ω = (0, π)n, for some n ∈ IN∗, and fix an arbitrary α ∈ (0, 1/2). According to
Theorem 4, there exists a unique optimal set, and moreover one has CT(χω∗) < CT,rand(χω∗).
According to [43, 44], the anomalous diffusion equation (7) is not exactly null controllable for α < 1/2, and therefore (by duality) CT(χω∗) = 0. Hence, we have here an example where
CT(χω∗) = 0 whereas C
T,rand(χω∗) > 0.
2.4.4 Several numerical simulations
We provide hereafter several numerical simulations, illustrating the above results. The truncated problem of order N is obtained by considering all couples (j, k) such that j 6 N and k 6 N . The simulations are made with a primal-dual approach combined with an interior point line search filter method7.
On Figure1(resp., on Figure2), we compute the optimal domain ωN for the operator A
0= −△,
the Dirichlet-Laplacian (resp., the Neumann-Laplacian on the domain defined with zero average) on the square Ω = (0, π)2. We can observe the expected stationarity property of the sequence of
optimal domains ωN from N = 4 on (i.e., 16 eigenmodes).
Note that, in the numerical simulations, we have taken T = 0.05, that is, a small value. Indeed, in accordance with Remark10, if we take T too large then the stationarity property is observed from N = 1 on, and then the numerical simulations are not very meaningful.
Figure 1: On this figure, Ω = (0, π)2, L = 0.2, T = 0.05, and A
0 is the Dirichlet-Laplacian. Row
1, from left to right: optimal domain ωN (in green) for N = 1, 2, 3. Row 2, from left to right:
optimal domain ωN (in green) for N = 4, 5, 6.
7
More precisely, we used the optimization routine IPOPT (see [63]) combined with the modeling language AMPL (see [18]) on a standard desktop machine.
Figure 2: On this figure, Ω = (0, π)2, L = 0.2, T = 0.05, and A
0= −△ is the Neumann-Laplacian
defined on the domain D(A0) = {y ∈ H2(Ω, C) | RΩy = 0 and ∂n∂y = 0 on ∂Ω}. Row 1, from left
to right: optimal domain ωN (in green) for N = 1, 2, 3. Row 2, from left to right: optimal domain
ωN (in green) for N = 4, 5, 6.
On Figures 3 and 4, we compute the optimal domain ωN for the operator A
0 = (−△)α, the
fractional Dirichlet-Laplacian on the unit disk Ω = {x ∈ IR2 | kxk < 1}, for α = 1 and α = 0.15. The numerical simulations illustrate the result stated in Theorem5. Indeed, in the case α = 1, we can observe the expected stationarity property of the sequence of optimal domains ωN from
N = 3 on (i.e., 9 eigenmodes). In the case α = 0.15, the numerical simulations provide evidence of the accumulation of concentric rings at the boundary (as expected); they are done with values of N between 1 and 15 (i.e., 225 eigenmodes).
Figure 3: On this figure, Ω = {x ∈ IR2 | kxk < 1}, L = 0.2, T = 0.05, and A0 is the
Dirichlet-Laplacian. Row 1, from left to right: optimal domain ωN (in green) for N = 1, 2, 3. Row 2, from
Figure 4: On this figure, Ω = {x ∈ IR2 | kxk < 1}, L = 0.2, T = 0.05, and A0 = (−△)α is the
fractional Dirichlet-Laplacian with α = 0.15. Row 1, from left to right: optimal domain ωN (in
green) for N = 1, 2, 5. Row 2, from left to right: optimal domain ωN (in green) for N = 10, 12, 15.
These figures show what must be the optimal shape and placement of a thermometer in a square domain or in a disk (for the corresponding boundary conditions), when the observation is made over the horizon of time [0, T ].
Remark 14. If α = 1 then we are in the framework of Theorem1 and hence N0(T ) = 1 if T is
large enough. With respect to what is drawn on Figure3, this means that if T is large enough then the optimal set is simply the central disk. The situation is however much more complicated if α < 1/2 (as on Figure 4), since it is proved that a finite number of modes is never sufficient in order to recover the optimal set. In that case, for every value of T the optimal set will always consist of an infinite number of concentric rings accumulating at the boundary, and it is an open and interesting question to investigate how the optimal set behaves when T tends to +∞.
2.5
Further comments from a semi-classical analysis viewpoint
The assumption (H2) is of a spectral nature and can be seen from a semi-classical analysis viewpoint
as follows. The probability measure µj = φj(x)2dx is interpreted (in quantum mechanics) as the
probability of being in the state φj with an energy λj. Every closure point or weak limit for
the vague topology of the sequence of probability measures (µj)j∈IN∗ is called a semi-classical
measure or a quantum limit (the general definition is however in the phase space). In this sense,
the assumption (H2) can be called a ”lower-bound semi-classical assumption”.
The question of determining the set of quantum limits is widely open in general. One is able to compute them only in very particular cases. In the standard round sphere (in any dimen-sion) any geodesic invariant measure is a quantum limit (see [32]), hence in particular the Dirac along any geodesic circle is a quantum limit. This provides an account for possible strong con-centrations of eigenfunctions. Similarly, in the disk with Dirichlet boundary conditions, the Dirac along the boundary is a quantum limit (accounting for the already mentioned whispering galleries phenomenon) In contrast, in the flat torus (in any dimension) all quantum limits are absolutely continuous (see [31]).
In some sense the assumption (H2) stipulates that there is no very strong concentration
phe-nomenon. To be more precise, we claim that:
The assumption (H2) holds true if one is able to establish that the eigenfunctions φj
are uniformly bounded in L∞ and that every semi-classical measure (weak limit of
the probability measures µj for the vague topology) is absolutely continuous and the
corresponding densities are positive over the whole domain Ω.
This claim easily follows from the Portmanteau theorem (see also Remark15further), because then, using the fact that γj(T ) is exponentially increasing, it follows that γj(T )RΩa(x)φj(x)2dx → +∞
for every a ∈ UL.
Unless the case of flat tori mentioned above, we are not aware of existing results establishing exactly such a property, however results in this direction can be found in [3, 12]. Note that this property holds true for square domains (as explained previously).
In general, there are many possible quantum limits. The most natural one is the uniform measure, and it is indeed an important issue in quantum physics is to determine appropriate assumptions on Ω under which the probability measures µj tend to equidistribute as j converges
to +∞. The famous Schnirelman theorem (see [15, 21, 25, 57, 66]) states that, if Ω is ergodic with a piecewise smooth boundary, then8 there exists a subsequence of (µ
j)j∈IN∗ of density one
converging vaguely to the uniform measure 1
|Ω|dx (Quantum Ergodicity on the base). Here, density
one means that there exists I ⊂ IN∗ such that #{j ∈ I | j 6 N}/N converges to 1 as N → +∞, and the manifold is seen as a billiard where the geodesic flow moves at unit speed and bounces at the boundary according to the Geometric Optics laws.
The Shnirelman theorem lets however open the possibility of having an exceptional sequence of measures µj converging vaguely, e.g., to an invariant measure carried by unstable closed geodesic
orbits or on some invariant tori formed by such orbits. This kind of semi-classical measure is referred to as a scar and accounts for an energy concentration phenomenon.
Then, with respect to our discussion concerning the validity of the assumption (H2), the worst
possible case is when there exist a quantum limit which is completely concentrated, such as a scar. In this sense, the assumption (H2) is a ”non-scarring” assumption.
Remark 15. In the claim above (and in Theorem1) we have assumed that the eigenfunctions are uniformly bounded in L∞(Ω). This strong assumption holds true in domains that are Cartesian
products of one-dimensional domains, but for example if Ω is a ball then the eigenfunctions of the Dirichlet-Laplacian are not uniformly bounded.
It is interesting to understand why we add the strong assumption of L∞uniform boundedness. It is needed in the application of the Portmanteau theorem, for the following reason. In semi-classical analysis the vague topology for measures is usually employed. Assuming that the quantum limits under consideration are absolutely continuous, the convergence in vague topology means that (up to subsequence) lim j→+∞ Z ω φ2jdx = Z ω φ2dx ∀ω measurable s.t. |∂ω| = 0,
that is, the convergence holds on every Jordan measurable set. In contrast, the convergence in L1
weak topology means that lim j→+∞ Z ω φ2jdx = Z ω φ2dx ∀ω measurable, 8
Note that the results established in these references are actually stronger and derive the QE property, not only ”on the base” (that is, in the configuration space Ω), but in the unit cotangent bundle S∗Ω of Ω, in the framework
of pseudo-differential operators. Here, we are concerned only with weak limits in Ω, and following [65] we use the wording ”on the base”.
that is, the convergence does hold true as well for those measurable subsets whose boundary has a positive measure. Both convergence properties do coincide as soon as we add the L∞boundedness
assumption. This explains why we added such a strong assumption. Indeed our aim is to be able to capture any possible measurable subset.
3
Proofs
This section is devoted to prove Proposition2, Theorems1,3,4and5, and finally (in this order), Theorem2.
3.1
Proof of Proposition
2
For every N ∈ IN∗, the functional JN defined by (22) on UL is extended to UL (see Remark6) by
setting JN(a) = inf 16j6Nγj(T ) Z Ωa(x)|φ j(x)|2dx, (34)
for every a ∈ UL. We consider the relaxed truncated problem
sup
a∈UL
JN(a). (35)
Using the same arguments as in the proof of Lemma1, it is clear that the problem (35) has at least one solution aN ∈ U
L. Let us prove that aN is the characteristic function of a set ωN such
that χωN ∈ UL. Define the simplex set
SN = n α = (αj)16j6N ∈ IRN+ N X j=1 αj= 1 o .
It follows from the Sion minimax theorem (see [59]) that
sup a∈UL min 16j6Nγj(T ) Z Ωa(x)|φ j(x)|2dx = max a∈UL min α∈SN Z Ω a(x) N X j=1 αjγj(T )|φj(x)|2dx = min α∈SN max a∈UL Z Ω a(x) N X j=1 αjγj(T )|φj(x)|2dx,
and that there exists αN ∈ S
N such that (aN, αN) is a saddle point of the functional
(a, α) ∈ UL× SN 7−→ N X j=1 αjγj(T ) Z Ωa(x)|φ j(x)|2dx.
Therefore, aN is solution of the optimal design problem
max a∈UL Z Ω a(x) N X j=1 αNj γj(T )|φj(x)|2dx.
Set ϕN(x) = PNj=1αNj γj(T )|φj(x)|2, for every x ∈ Ω. It follows from (H1) that ϕN is never
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